Molecular Evolution
Inken Wohlers
Abstract
If we look at living organisms we find an amazingly complex system of
interacting molecules. We find that each organism’s genetic information is
encoded in RNA or DNA sequences out of four nucleotides and that each
organism proliferates by synthesis of new and basically identical RNA or
DNA sequences. A sequence’s genetic information instructs the synthesis
of a large number of different molecules which fulfill complex functions.
It is not known yet, how this complex molecular machinery we encounter
in living organisms has developed, but two molecular prerequisites are
obvious: self-reproduction and cooperation of (macro-)molecules. The
possibility to store more and more information and to be part of more and
more complex interactions seems to be a result of mutation and selection
and thus evolution on a molecular level. Eigen and Schuster proposed two
mechanisms for such a molecular evolution, quasi species and hypercycles
[1]. The quasi species model explains the inner diversity of species while
hypercycles explain how the genetic information stored in an RNA or
DNA sequence can be increased as a result of molecular cooperation.
1
Quasi Species
Manfred Eigen and Peter Schuster proposed a model for the evolution of selfreplicating molecules like RNA or DNA, the quasi species model. An outline of
this model will be given in chapter 1. For more details and the mathematical
formulation refer to a review by Kemena [2] or the original publication [1]. The
limitations of the quasi species model will be discussed in chapter 2. To overcome these limitations Eigen and Schuster suggest hypercycles, which will be
introduced in chapter 3.
A self-replicating molecule consists out of a sequence of monomers and results
from a copy process of an existing molecule. It can be identical with its parent or
can be mutated from it leading to a locally different molecule. A self-replicating
molecule may decay into its building blocks, in the case of RNA or DNA into
nucleotides. The model assumes that the building blocks are always present in
sufficient amount. The selective pressure on a specific self-replicating molecule
is its replication rate; molecules with higher replication rate have a selective advantage since they are able to reproduce themselves or, as a result of mutations,
sequences closely related to themselves more often.
The main finding of the mathematical formulation of this model is the existence
of so called quasi species, which are distributions of closely related sequences.
The relatedness is usually measured by the minimal number of point mutations
needed in order to transform one sequence in the other.
The selective pressure does not lead to survival of one single sequence, but to
1
consolidation of many different sequences which differ only slightly from each
other and thus can mutate into each other during replication. The sequence
with the highest number of molecules is called the master sequence and refers
to what is called the wildtype. The quasi species is distributed around this
master sequence where the form of the distribution may vary. If the mutation
rate is small less mutations are likely to occur and the quasi species distribution
gets narrow, while a high mutation rate leads to more mutations resulting in a
broader distribution.
Figure 1: Quasi species in the room of sequences [3].
There are two borderline cases for the mutation rate. The first one is a
mutation rate of 0, which means, that each sequence is copied without error,
always leading to an identical sequence. In this case each sequence represents
a quasi species. The other borderline case is that the mutation rate exceeds
a specific error threshold which leads to random sequence spreading over the
whole range of possible sequences. The quasi species distribution is unsteady
or randomly distributed and its information therefore can not be passed on
through various reproductions. As a result information can not be inherited.
The next chapter describes in detail how the error threshold can be obtained
and how this leads to a limit for the information which can be conserved within
a quasi species.
2
Information Content and Conservation of Information
A quasi species is able to pass on information to consecutive generations if its
distribution is steady within the room of sequences. This is the case if and only
if the master sequence is stable. In order to derive a formula for the error rate
threshold we have a look at the probability for an error free replication of the
master sequence, Qm . The probability of a mutation follows to be 1 − Qm . Qm
can be interpreted as a quality factor for the replication and can be written as
product of the probabilities for error-free replication of the single monomers in
the master sequence. If we assume as a simplification that all nucleotides have
2
the same probability for error-free replication, namely qm , we get
Nm
Qm = qm
,
(1)
where Nm is the length of the master sequence. The master sequence is stable if
the correct master sequence copies are able to compete with their error copies,
which means that the quality factor needs to be above a certain threshold,
formally
1 > Qm > Qmin .
(2)
For the threshold Qmin the following equation can be derived
Qm > Qmin =
1
,
σm
(3)
which is equivalent to
r
qmin =
Nm
1
,
σm
(4)
where σm is the superiority of the master sequence. The superiority is a weighted
quotient of the master sequence’s fitness divided by the rest of the population’s
fitness. Equation (4) shows that the probability for an error-free replication
and therefore also the error rate threshold depends on two parameters, on the
superiority of the master sequence and on the sequence length. From rearranging equation (4) we obtain an equation for the maximum sequence length
that still permits a stable quasi species distribution and therefore inheritance of
information,
ln σm
Nmax =
.
(5)
1 − qm
Note that this limit is inversely proportional to the average error rate per
monomer within the master sequence, 1−qm . Another way to interpret equation
(5) is as expectation value of an error in the master sequence,
E(εm ) = Nm (1 − qm ) ⇔ exp(E(εm )) < σm .
(6)
The expected number of mutations during one reproduction of the master sequence has to stay below a sharply defined threshold which depends on the
superiority of the master sequence. The relationship of sequence length, error rate and superiority described by equation (5) is illustrated in table 2 for
different molecular mechanisms.
3
Hypercycles
The last chapter showed that the information content of quasi species is strictly
limited. In table 2 sequence lengths for enzyme-free RNA replication are listed
for different values of superiority. Uncatalyzed replication of RNA has never
been observed to any satisfactory extent but error rates of 5×10−2 per nucleotide
seem to be realistic. For such fairly high error rates sequence lengths up to a
magnitude of 100 base pairs can be reached for high values of superiority. Such
low information content is not sufficient to encode for the complexity of living
systems we encounter today. But how can the sequence length and thus the
information content of a system be increased? Eigen and Schuster proposed
3
Error
Rate
1 − qm
Superiority
σm
Sequence
Length
Nmax
Molecular Mechanism
and Biological Example
5 × 10−2
2
20
200
14
60
106
5 × 10−4
2
20
200
1386
5991
10597
1 × 10−6
2
20
200
0.7 × 106
3.0 × 106
5.3 × 106
DNA replication via polymerases
including proofreading by exonuclease
(E.coli, N = 4 × 106 )
1 × 10−9
2
20
200
0.7 × 109
3.0 × 109
5.3 × 109
DNA replication and
recombination in eucaryotic cells
(vertebrates (man), N = 3 × 109 )
Enzyme-free RNA replication
(t-RNA precursor, N = 80)
Single-stranded RNA
replication via specific replicases
(phage Qβ , N = 4500)
Figure 2: Error rate, superiority and sequence length for different molecular
mechanisms [1].
4
cooperation of macromolecules within a hypercyclic linkage as an answer to this
question. This chapter will discuss their model of the hypercycle in detail.
From equation (5) we know that there are two ways to increase the sequence
length of a self-replicating molecule and thus the information content of its quasi
species. Those are:
• Lower the error rate
• Increase the master sequence’s superiority
Both can be achieved via cooperation of molecules. In the case of RNA or DNA
replication for example enzymes catalyze the replication and therefore lower the
error rate. Lower error rates permit for larger sequence lengths. The enlarged
information content can encode more complex enzymes and therefore increase
the superiority. This is visualized in figure (3).
Figure 3: Relationship of RNA/DNA sequences and enzymes.
Hypercycles are chemical reaction cycles of a high level of organization. A
chemical reaction cycle is a sequence of reactions where any product is identical
with a reactant of a preceding step. Chemical reaction cycles are also called
catalysts because they catalyze the synthesis of at least one member of the
cycle. Figure (4) shows the catalytic mechanism of an enzyme.
Figure 4: Catalyst: Catalytic mechanism of an enzyme (S: Substrate, P: Product, E: Enzyme).
The next higher level of organization is the catalytic cycle, a reaction cycle
where one or up to all intermediates are catalysts. An example for a catalytic
cycle is the replication of single stranded RNA which exhibits linear growth
behavior since from one single strand one new strand is produced. The abstract
catalytic cycle and the catalytic cycle for single stranded RNA replication is
displayed in figure (5).
An autocatalyst is a special catalytic cycle where the product of a single
reaction is identical with its substrate. Therefore it is a true self-replicating
5
Figure 5: Catalytic cycles. Left: Schema where Ei are enzymes and S substrate.
Right: Replication of single-stranded RNA.
unit. An example for an autocatalytic cycle is the process of semi-conservative
DNA replication where two daughter strands are synthesized from one parent
strand. Autocatalysts exhibit exponential growth behavior.
Figure 6: Autocatalyst: Left: Schema where E is the enzyme and S the substrate. Right: Semi-conservative DNA replication.
A hypercycle finally is a reaction cycle of a next higher level of organization,
namely a cycle of catalytic cycles or autocatalysts. In the case of RNA or DNA
molecules we find hypercyclic cooperation if the sequences are transcribed and
translated to enzymes, which in return catalyze the replication of the RNA or
DNA sequences and therefore lower the replication’s error rate. The abstract
form of a hypercycle is displayed in figure (7).
If we assume that self-organization within hypercycles was one step of molecular evolution, we might ask ourselves, why nowadays there exists only one
hypercycle for the molecular machinery of the cell and why there is only one
universal genetic code. This is the reason, why we finally have a look at the
competition of hypercycles.
Hypercycles compete with each other if they are not locally separated from each
other, if they need the same chemical building blocks and if they do not cooperate in higher order linkage. The number of molecules for each hypercycle can
6
Figure 7: Hypercycle: Ii are catalytic cycles or autocatalysts, Ei are enzymes.
be mold using differential equations [4],
dnE,i
= AT,i nI,i − ϕE (t)nE,i
(7)
dt
dnI,i
= AR,i nI,i nE,i − ϕI (t)nI,i .
(8)
dt
Here nE,i is the number of enzymes and nI,i the number of self-replicating
molecules, e.g. RNA or DNA. The parameters are AT for the rate with which
a self-replicating molecule is translated to an enzyme and AR for its replication
rate. The index i denotes the hypercycle the molecules and rate constants
belong to. The change of the number of enzymes depends on the number of
self-replicating molecules present. The change of the number of self-replicating
molecules depends on the number of enzymes catalyzing and on the number of
self-replicating molecules since they are the blueprint for new molecules. The
competition of hypercycles is mold using the functions ϕE (t) and ϕI (t) which
depend on the time t. The competition therefore changes with time. As a
simplification we assume that the overall number of molecules in the system is
constant. This means that the rates of change, the values of the differential
equations, summed up have to be equal to zero, formally
X dnI,i
X dnE,i
=0,
= 0.
(9)
dt
dt
i
i
Using those equations we get for the competition functions
P
P
AR,i nI,i nE,i
i AT,i nI,i
P
ϕE =
, ϕI = i P
.
n
k E,k
k nI,k
(10)
Now we are interested in the stable steady states of this dynamic system, since
stable steady states are specific numbers of molecules that are reached after
a certain time and that are maintained, even after small perturbations. If we
consider the competition of two hypercycles we find two stable steady states,
(nE,1 , nI,1 , nE,2 , nI,2 ) = (0, 0, CE , CI )
(11)
(nE,1 , nI,1 , nE,2 , nI,2 ) = (CE , CI , 0, 0),
(12)
7
where CE = nE,1 + nE,2 is the overall number of enzymes and CI = nI,1 + nI,2
the overall number of self-replicating molecules in the system. The trajectories
in the (nI,1 , nE,1 ) plane are depicted in figure (8).
Figure 8: Competition of two hypercycles: Nullclines (blue and green lines)
and trajectories for the first hypercycle (white lines) (AT,1 = AR,1 = 1, AT,2 =
AR,2 = 0.5)
The steady states and the trajectories show that depending on the initial
values only one of the two hypercycles survives. In the case which the graphic visualizes it is less likely that the first hypercycle with better replication and translation rates gets extinct, but it may happen if the initial number of molecules
belonging to this hypercycle is small. Since a new evolving hypercycle usually
has a low initial number of molecules it is unlikely that it can survive if another
hypercycle is already present, even if the new hypercycle has a better reproduction rate. This fact can be described as ”once-for-ever” selection [1]: After
one hypercycle had been selected and gained many molecules, there is almost
no chance for new hypercycles to prevail, even if they are more efficient. Such
a ”once-for-ever” selection is a possible explanation why there is only one basic
molecular machinery of the cell.
References
[1] Schuster P Eigen M. The hypercycle: A principle of natural selforganization. Naturwissenschaften, 64:541–565, 1977.
8
[2] ’Carsten Kemena’. Sequence evolution. Seminar ’Gute Ideen in der theoretischen Biologie/Systembiologie’ FU Berlin, 2007.
[3] Schuster P.
Ursprung des lebens und evolution von moleklen.
http://www.tbi.univie.ac.at/ pks/PublicLectures/ursprung-1999.pdf, 1999.
[4] Fachkurs ’Modellierung biologischer Systeme’. Competition and selection in
biological systems. HU, 2000.
9